Efficient Quantum Compiling

Efficient quantum compiling is a necessary tool for future quantum computers. We have developed an efficient classical algorithm to compile an arbitrary unitary operator into a product of Clifford+T operators, which can be implemented fault-tolerantly on a quantum computer. Runtime and product length for our algorithm are $O(4^NN\log(1/\epsilon))$, where $N$ is the number of qubits and $\epsilon$ is the error (normalized trace distance); these values are equivalent to the current state of the art. A theoretical lower bound for both values is $\Omega(4^N\log(1/\epsilon))$.